Let the space of probabilities \(\rho (x)\)  be \(\mathcal {E}_{\rho }\)  and the space of phase functions \(\Phi (x)\)  be \(\mathcal {E}_{\Phi }\)  so that the joint space can be written asWe are not concerned here with the detailed structure of \(\mathcal {E}_{\rho }\)  and \(\mathcal {E}_{\Phi }\) , however, it is interesting to note that the space \(\mathcal {E}_{\rho }\)  does have some interesting properties. For one, it has a Riemannian metric, the Fisher-Rao metric,
\(dl^{2} = C \int dx\frac{\delta \rho _{x}^{2}}{\rho _{x}} = 4C\int dx \left(\delta \rho ^{1/2}_{x}\right)^{2}~.\) 
This structure turns out to be valuable for reconstructing the full formalism of quantum theory (see e.g., [1]}).

\(\Gamma = \mathcal {E}_{\rho }\otimes \mathcal {E}_{\Phi }.\) 
